How can I connect two seams on a mesh? They come apart while posing but not while modeling
Idle Chit-chat

Ok, I switched to answer to give a more detailed, well, answer! It appears that your mesh is not actually one piece, despite appearances.

Problem

I am not familiar with the self-intersect tool you described (or Ctrl + F if those are one and the same), but the spheres are separate. I proved this by selecting a single verticy of one of the spheres and keying L. If the mesh were a whole, like rigging requires, everything would select. Everything, obviously, did not.

Solution

Luckily, there is a simple solution! Select the entire mesh with A, key W, and select Remove Doubles:



This will join all the seemingly joined edge loops. Now you can continue on and rig!